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is Lebesgue null and has Jordan outer content one
Statement refuted
A Lebesgue null subset of must have Jordan outer content .
Facts & Assumptions
Given: The Axiom of Countable Choice and the set .
Every at most countable subset of is Lebesgue null (Every at most countable subset of is Lebesgue null; in particular ).
Its Jordan outer content is the infimum of over finite axis-parallel rectangle covers of the set (Jordan inner and outer content and Jordan measurable bounded sets in ).
The same lower bound on total length holds for a finite cover of an interval by bounded intervals of any of the four bounded forms (If finitely many intervals cover a closed bounded interval , the sum of their lengths is at least ).
The rationals are countably infinite ( is countably infinite).
Counterexample
The witness set is countable, hence Lebesgue null by [L1].
Let finitely many bounded intervals cover . Their union is closed in , and because it contains the dense subset of , [F3] makes it contain all of .
Therefore [F2] gives total covering length at least .
The single interval realises total length , so [F1] gives Jordan outer content exactly . Thus is Lebesgue null and still has Jordan outer content one.
Depends on
- Every at most countable subset of $\mathbb{R}^n$ is Lebesgue null; in particular $\lambda_1(\mathbb{Q})=0$
- Jordan inner and outer content and Jordan measurable bounded sets in $\mathbb{R}^m$
- If finitely many intervals cover a closed bounded interval $[a,b]$, the sum of their lengths is at least $b - a$
- $\mathbb{Q}$ is countably infinite
- Both $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ are dense in $\mathbb{R}$, and every nonempty open subset of $\mathbb{R}$ is uncountable
- Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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Sources
- T. Tao, An Introduction to Measure Theory (GSM 126), Exercise 1.2.1 (standard reference, not scraped)